Componentwise concave copulas and their asymmetry

summary:The class of componentwise concave copulas is considered, with particular emphasis on its closure under some constructions of copulas (e.g., ordinal sum) and its relations with other classes of copulas characterized by some notions of concavity and/or convexity. Then, a sharp upper bound is given for the $L^{\infty}$-measure of non-exchangeability for copulas belonging to this class

Constructing copulas from shock models with imprecise distributions

The omnipotence of copulas when modeling dependence given marg\-inal distributions in a multivariate stochastic situation is assured by the Sklar's theorem. Montes et al.\ (2015) suggest the notion of what they call an \emph{imprecise copula} that brings some of its power in bivariate case to the imprecise setting. When there is imprecision about the marginals, one can model the available information by means of $p$-boxes, that are pairs of ordered distribution functions. By analogy they introduce pairs of bivariate functions satisfying certain conditions. In this paper we introduce the imprecise versions of some classes of copulas emerging from shock models that are important in applications. The so obtained pairs of functions are not only imprecise copulas but satisfy an even stronger condition. The fact that this condition really is stronger is shown in Omladi\v{c} and Stopar (2019) thus raising the importance of our results. The main technical difficulty in developing our imprecise copulas lies in introducing an appropriate stochastic order on these bivariate objects

Transformations of Copulas and Measures of Concordance

Copulas are real functions representing the dependence structure of the distribution of a random vector, and measures of concordance associate with every copula a numerical value in order to allow for the comparison of different degrees of dependence. We first introduce and study a group of transformations mapping the collection of all copulas of fixed but arbitrary dimension into itself. These transformations may be used to construct new copulas from a given one or to prove that certain real functions on the unit cube are indeed copulas. It turns out that certain transformations of a symmetric copula may be asymmetric, and vice versa. Applying this group, we then propose a concise definition of a measure of concordance for copulas. This definition, in which the properties of a measure of concordance are defined in terms of two particular subgroups of the group, provides an easy access to the investigation of invariance properties of a measure of concordance. In particular, it turns out that for copulas which are invariant under a certain subgroup the value of every measure of concordance is equal to zero. We also show that the collections of all transformations which preserve symmetry or the concordance order or the value of every measure of concordance each form a subgroup and that these three subgroups are identical. Finally, we discuss a class of measures of concordance in which every element is defined as the expectation with respect to the probability measure induced by a fixed copula having an invariance property with respect to two subgroups of the group. This class is rich and includes the well-known examples Spearman's rho and Gini's gamma

Actuarial Ratemaking in Agricultural Insurance

A scientific agricultural (re)insurance pricing approach is essential for maintaining sustainable and viable risk management solutions for different stakeholders including farmers, governments, insurers, and reinsurers. The major objective of this thesis is to investigate high dimensional solutions to refine the agricultural insurance and reinsurance pricing. In doing so, this thesis develops and evaluates three high dimensional approaches for constructing actuarial ratemaking framework for agricultural insurance and reinsurance, including credibility approach, high dimensional copula approach, and multivariate weighted distribution approach. This thesis comprehensively examines the ratemaking process, including reviews of different detrending methods and the generating process of the historical loss cost ratio's (LCR's, which is defined as the ratio of indemnities to liabilities). A modified credibility approach is developed based on the Erlang mixture distribution and the liability weighted LCR. In the empirical analysis, a comprehensive data set representing the entire crop insurance sector in Canada is used to show that the Erlang mixture distribution captures the tails of the data more accurately compared to conventional distributions. Further, the heterogeneous credibility premium based on the liability weighted LCR’s is more conservative, and provides a more scientific approach to enhance the reinsurance pricing. The agriculture sector relies substantially on insurance and reinsurance as a mechanism to spread loss. Climate change may lead to an increase in the frequency and severity of spatially correlated weather events, which could lead to an increase in insurance costs, or even the unavailability of crop insurance in some situations. This could have a profound impact on crop output, prices, and ultimately the ability to feed the world rowing population into the future. This thesis proposes a new reinsurance pricing framework, including a new crop yield forecasting model that integrates weather and crop production information from different risk geographically related regions, and closed form reinsurance pricing formulas. The framework is empirically analyzed, with an original weather index system we set up, and algorithms that combine screening regression (SR), cross validation (CV) and principle component analysis (PCA) to achieve efficient dimension reduction and model selection. Empirical results show that the new forecasting model has improved both in-sample and out-of-sample forecasting abilities. Based on this framework, weather risk management strategies are provided for agricultural reinsurers. Adverse weather related risk is a main source of crop production loss, and in addition to farmers, this exposure is a major concern to insurers and reinsurers who act as weather risk underwriters. To date, weather hedging has had limited success, largely due to challenges regarding basis risk. Therefore, this thesis develops and compares different weather risk hedging strategies for agricultural insurers and reinsurers, through investigating the spatial dependence and aggregation level of systemic weather risks across a country. In order to reduce basis risk and improve the efficiency of weather hedging strategies, this thesis refines the weather variable modeling by proposing a flexible time series model that assumes a general hyperbolic (GH) family for the margins to capture the heavy-tail property of the data, together with the Lévy subordinated hierarchical Archimedean copula (LSHAC) model to overcome the challenge of high-dimensionality in modeling the dependence of weather risk. Wavelet analysis is employed to study the detailed characteristics within the data from both time and frequency scales. Results show that it is of great importance of capturing the appropriate dependence structure of weather risk. Further, the results reveal significant geographical aggregation benefits in weather risk hedging, which means that more effective hedging may be achieved as the spatial aggregation level increases. It has been discussed that it is necessary to integrate auxiliary variables such as weather, soil, and other information into the ratemaking system to refine the pricing framework. In order to investigate a possible scientific way to reweight historical loss data with auxiliary variables, this thesis proposes a new premium principle based on multivariate weighted distribution. Some designable properties such as linearity and stochastic order preserving are derived for the new proposed multivariate weighted premium principle. Empirical analysis using a unique data set of the reinsurance experience in Manitoba from 2001 to 2011 compares different premium principles and shows that integrating auxiliary variables such as liability and economic factors into the pricing framework will redistribute premium rates by assigning higher loadings to more risky reinsurance contracts, and hence help reinsurers achieve more sustainable profits in the long term